This article is cited in 12 papers

Short Communications

On the maximum of a Gaussian stationary process

M. G. Šur

Moscow

Abstract: We consider a Gaussian real process $x(t)$ which satisfies the same conditions as in [1]. We prove the existence (a.s.) of such random number $t_0$ ($t_0<\infty$) that the inequality
$$ |\max_{o\le u\le t}x(u)-\sigma\sqrt{2\ln t}|<\frac{(\sigma+\varepsilon)\ln\ln t}{\sqrt{2\ln t}} $$
is valid for all $t>t_0$ where $\varepsilon$ is any fixed positive number and $\sigma^2=\mathbf Mx^2(t)$.

Received: 07.10.1964

 English version: Theory of Probability and its Applications, 1965, 10:2, 354–357

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